Question $1$

Marks

Consider a liquid in a cylindrical container in which both the container and the liquid are rotating as a rigid body. The elevation difference $h,$ between the centre of the liquid surface and the rim of the liquid surface is a function of rotational speed $,$ fluid density $,$ gravitational acceleration vec$(g),$ and radius R $.$

$($a$)$ Use the Buckingham Pi method to find a dimensionless relationship between the parameters. Show all workings.

$[15]$

$($b$)$ State and explain two similarity rules that must be adhered to when scaling the dimensionless groups obtained.

$[6]$

$($c$)$ Determine how the rotational speed $,$ should be scaled such that we can maintain similar operating conditions when $($i$)$ doubling the fluid density $,$ or $($ii$)$ doubling the container radius R $.$

$6$

$[$Total $25]$